Duhem, Quine and the other dogma

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14 Alexander Afriat Not only does tertium non datur not hold in physics, the possibilities of negation are limitless: the negation ¬H of hypothesis H can suggest, say, another hypothesis H ′ = ¬H; but why not some other H ′′ = ¬H or H ′′′ = ¬H or who knows what else. So even if it were possible to refute a hypothesis in physics, its refutation would certainly not lead to the confirmation of another hypothesis—whereas the rejection of a hypothesis in mathematics typically allows a single, definite conclusion to be reached. La contradiction expérimentale n’a pas, comme la réduction à l’absurde employée par les géomètres, le pouvoir de transformer une hypothèse physique en une vérité incontestable ; pour le lui conférer, il faudrait énumerer complètement les diverses hypothèses auxquelles un groupe déterminé de phénomènes peut donner lieu ; or, le physicien n’est jamais sur d’avoir épuisé toutes les suppositions imaginables ; la vérité d’une théorie physique ne se décide pas à croix ou pile. So Duhem’s rejection (¬ dogma2) of crucial experiments turns on a ‘cleavage’ which resembles the one (dogma1) repudiated in “Two dogmas,” where it is claimed the dogmas are “two sides of a single dubious coin” (dogma1 ⇔ dogma2). Since the holism Duhem dwells on in Ch. VI §II (Qu’une expérience en Physique ne peut jamais condamner une hypothèse isolée, mais seulement tout un ensemble théorique) appears to be largely responsible for the cleavage invoked repeatedly in the following section, §III (L’experimentum crucis est impossible en physique), it could seem that overcoming holism would undermine that cleavage. This brings us to the difficulty raised at the beginning: that holism appears to have conflicting implications for Duhem and for Quine. In this connection let us briefly consider relations between Duhem’s §II and §III (Ch. VI). One relation is immediate succession—§III comes right after §II; another is that both are about crucial experiments. §II explains how holism prevents experiments from being crucial, the next section directly relates the impossibility of crucial experiments to the cleavage dividing physics and mathematics; one almost sees a simple syllogism: II Holism prevents experiments from being crucial. III The impossibility of crucial experiments makes physics unlike mathematics. ∴ Holism makes physics unlike mathematics. The trouble is that the differences between physics and mathematics are only partly due to holism; single-valued, invertible negation, 28 for instance, which holds in mathematics but not in physics according to Duhem, has little to do with holism. 28 One can write ¬(¬H) = H.
Duhem, Quine and the other dogma 15 Holism, which for Quine undermines meaning and hence analyticity, is therefore not entirely responsible for the cleavage repeatedly invoked by Duhem in his rejection of crucial experiments. It must also be said that mathematics may not be as analytic as I have taken it to be; Kant and others have regarded much of it as synthetic. Kant (1787 B190) defines the synthetic in terms of the principle of contradiction: Der Satz nun: Keinem Dinge kommt ein Prädikat zu, welches ihm widerspricht, heißt der Satz des Widerspruchs [. . . ]. Denn, wenn das Urteil analytisch ist, es mag nun verneinend oder bejahend sein, so muß dessen Wahrheit jederzeit nach dem Satze des Widerspruchs hinreichend können erkannt werden. 29 And Poincaré ([41] Ch. I) writes that the règle du raisonnement par récurrence— which he considers the raisonnement mathématique par excellence—is irréductible (involving infinitely many syllogisms) au principe de contradiction, and hence is the véritable type du jugement synthétique a priori. Crowe [20] argues that mathematics shares many of the difficulties attributed to physics in La théorie physique, and that Duhem attaches such weight to the distinctions of §III out of ignorance that mathematics is not so certain and ‘analytic’ after all. Physics, by becoming more and more detached from the world, seems moreover to be losing its synthetic character, and may have begun decades ago. The association of mathematics with the analytic, physics with the synthetic, could therefore be less straightforward than I have made it out to be. But again, definite resolution has not been my purpose; I have rather tried to explore the web of issues involved, and view in a fresh—perhaps questionable—light. 8 Final remarks Troubling shades of grey have prevailed in these pages over the reassuring certainties of black and white. I have often spoken of degree and nuance, of more and less, rather than of sic et non, of true and false: holism is undermined but not completely eradicated, meaning acquires much definiteness, analyticity is recovered to the extent that holism is overcome and so on. But isn’t Quine’s point that analytic and synthetic differ in degree and not in kind The gains in cruciality and analyticity with respect to the concerns of Duhem and Quine may be a matter of degree, but that degree seems considerable, perhaps considerable enough to warrant representation as promotions ‘in kind.’ Indeed it can be misleading not to view certain differences in degree as differences in kind—and hence, for instance, not to call the unlikeliest events ‘impossible,’ to 29 Translation: “Now the statement that nothing can have a predicate which contradicts it, is called the principle of contradiction [. . . ]. For if the judgement is analytic—be it negative or affirmative—its truth must always be adequately recognised by means of the principle of contradiction.”
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Duhem, Quine and the other dogma

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